MinT - Best Known (128−51, 128, s)-Nets in Base 8

Best Known (128−51, 128, s)-Nets in Base 8

(128−51, 128, 354)-Net over F₈ — Constructive and digital

Digital (77, 128, 354)-net over F₈, using

12 times m-reduction [i] based on digital (77, 140, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 70, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(128−51, 128, 384)-Net in Base 8 — Constructive

(77, 128, 384)-net in base 8, using

8² times duplication [i] based on (75, 126, 384)-net in base 8, using
- trace code for nets [i] based on (12, 63, 192)-net in base 64, using
  - base change [i] based on digital (3, 54, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(128−51, 128, 572)-Net over F₈ — Digital

Digital (77, 128, 572)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−51, 128, 56247)-Net in Base 8 — Upper bound on s

There is no (77, 128, 56248)-net in base 8, because

1 times m-reduction [i] would yield (77, 127, 56248)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 925921 334778 209476 351811 173182 834652 744876 779991 634815 370389 585958 059187 426091 363954 647288 963048 105909 705669 189476 > 8¹²⁷ [i]